Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2006  June  Q#9
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Question
Given that f(x)=( x^{2} – 6x)(x – 2) + 3x,
a. express f(x) in the form x(ax^{2} + bx + c), where a, b and c are constants.
b. Hence factorise f(x) completely.
c. Sketch the graph of y = f(x), showing the coordinates of each point at which the graph meets the axes.
Solution
a.
We are given;
b.
c.
We are required to sketch
We have seen in (a) that it can written as;
It is a cubic function.
ü Find the sign of the coefficient of
ü Find the point where the graph crosses yaxis by finding the value of
ü Find the point(s) where the graph crosses the xaxis by finding the value of
ü Calculate the values of
ü Complete the sketch of the graph by joining the sections.
Sketch should show the main features of the graph and also, where possible, values where the graph intersects coordinate axes.
First we find xintercepts.
The point
We have found in (b) that given cubic function can be written as;
Therefore;
Now we have three options.








Therefore, the graph intersects xaxis at three points (0,0), (3,0) and (5,0).
Next we find yintercept.
The point
Therefore;
Hence, the graph intercepts yaxis at (0,0).
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